**Introduction to Matrices**

**What is a Matrix?**

A matrix is a rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.

**History of the Matrix**

The matrix has a long history of application in solving linear equations. They were known as arrays until the 1800‘s. The term “matrix” (Latin for “womb”, derived from *mater*—mother) was coined by James Joseph Sylvester in 1850, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that are derived from the original one by removing columns and rows. An English mathematician named Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation *A*=*ai*,*j* to represent a matrix where *ai*,*j* refers to the element found in the *i*th row and the *j*th column. Matrices can be used to compactly write and work with multiple linear equations, referred to as a system of linear equations, simultaneously. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps.

**What is a Matrix**

In mathematics, a *matrix* (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are commonly written in box brackets. The horizontal and vertical lines of entries in a matrix are called *rows* and *columns*, respectively. The size of a matrix is defined by the number of rows and columns that it contains. A matrix with *m* rows and *n* columns is called an *m* × *n* matrix or *m*-by-*n* matrix, while *m* and *n* are called its *dimensions*.The dimensions of the following matrix are 2×3 up(read “two by three”), because there are two rows and three columns.

*A*=[19−13205−6]

Matrix Dimensions: Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A.

The individual items (numbers, symbols or expressions) in a matrix are called its *elements* or *entries*.

Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field.

Matrices which have a single row are called *row vectors*, and those which have a single column are called *column vectors*. A matrix which has the same number of rows and columns is called a *square* *matrix*. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an *empty matrix*.

**Addition and Subtraction; Scalar Multiplication**

Matrix addition, subtraction, and scalar multiplication are types of operations that can be applied to modify matrices.

There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication. These form the basic techniques to work with matrices.

These techniques can be used in calculating sums, differences and products of information such as sodas that come in three different flavors: apple, orange, and strawberry and two different packaging: bottle and can. Two tables summarizing the total sales between last month and this month are written to illustrate the amounts. Matrix addition, subtraction and scalar multiplication can be used to find such things as: the sales of last month and the sales of this month, the average sales for each flavor and packaging of soda in the 22-month period.

**Adding and Subtracting Matrices**

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. Matrix addition is commutative and is also associative, so the following is true:

A+B=B+AA+B=B+A

(A+B)+C=A+(B+C)(A+B)+C=A+(B+C)

**Scalar Multiplication**

In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. What does it mean to multiply a number by 33? It means you add the number to itself 33 times. Multiplying a matrix by 33 means the same thing; you add the matrix to itself 33 times, or simply multiply each element by that constant.